$\int \sqrt{x^{2} + 3 x} dx$ is equal to

Moderate

A

$\frac{2 x + 3}{2} \sqrt{x^{2} + 3 x} - \frac{9}{4} \log (\frac{2 x + 3}{2}) + \sqrt{x^{2} + 3 x} ∣ + C$
B

$\frac{2 x + 3}{4} \sqrt{x^{2} + 3 x} - \frac{9}{4} \log (\frac{2 x + 3}{2}) + \sqrt{x^{2} + 3 x} ∣ + C$
C

$(x + \frac{3}{2}) \sqrt{x^{2} + 3 x} - \frac{9}{8} \log (x + \frac{3}{2}) + \sqrt{x^{2} + 3 x} ∣ + C$
D

None of the above

Solution

Let
$\begin{matrix} I = \int \sqrt{x^{2} + 3 x} dx = \int \sqrt{x^{2} + 3 x - {(\frac{3}{2})}^{2} + {(\frac{3}{2})}^{2}} dx \\ = \int \sqrt{{(x + \frac{3}{2})}^{2} - {(\frac{3}{2})}^{2}} dx \\ I = \frac{x + \frac{3}{2}}{2} \sqrt{{(x + \frac{3}{2})}^{2} - {(\frac{3}{2})}^{2}} - \frac{9}{4 \times 2} \log ((x + \frac{3}{2}) + \sqrt{x^{2} + 3 x} ∣ + C \end{matrix}$
$\begin{aligned} [∵ \int \sqrt{x^{2} - a^{2}} dx = \frac{x}{2} \sqrt{x^{2} - a^{2}} - \frac{a^{2}}{2} \log |x + \sqrt{x^{2} - a^{2}}|] \\ \Rightarrow I = \frac{2 x + 3}{4} \sqrt{x^{2} + 3 x} - \frac{9}{8} \log |(\frac{2 x + 3}{2}) + \sqrt{x^{2} + 3 x}| + C \end{aligned}$

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